simplifying trig identity

Hi. looking for help with simplifying an identity. The identity to be simplified is:
tan^4x + 2tan^2x + 1.

I'm confident that 2tan^2x + 1 will break down into:
tan^2x + tan^2x + 1 which equals tan^2x + sec^2x. But I'm not sure how to treat the tan^4 when converting to secant.

Any help will be much appreciate.
Mike Clemmons

Quote:

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Originally Posted by Mike Clemmons

Hi. looking for help with simplifying an identity. The identity to be simplified is:
tan^4x + 2tan^2x + 1.

I'm confident that 2tan^2x + 1 will break down into:
tan^2x + tan^2x + 1 which equals tan^2x + sec^2x. But I'm not sure how to treat the tan^4 when converting to secant.

Any help will be much appreciate.
Mike Clemmons

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Note that the problem is in the form

$\displaystyle y^4+2y^2+1=(y^2+1)^2$

where $\displaystyle y=\tan(x)$

This should get you started.

Also recall that $\displaystyle \tan^2{\theta} + 1 = \sec^2{\theta}$.